On the large time behavior of heat kernels on Lie groups

Noël Lohoué and Georgios Alexopoulos

Source: Duke Math. J. Volume 120, Number 2 (2003), 311-351.

Abstract

We prove Gaussian estimates for heat kernels on semisimple Lie groups by using the method of Bloch wave representation. We also give a large time asymptotic expansion for heat kernels on compact extensions of abelian Lie groups.

Primary Subjects: 22E80

Secondary Subjects: 43A90, 60B99, 60J60

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.dmj/1082138586
Digital Object Identifier: doi:10.1215/S0012-7094-03-12024-4
Zentralblatt MATH identifier: 1040.22004
Mathematical Reviews number (MathSciNet): MR2019978

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References

G. K. Alexopoulos, Sous-laplaciens et densités centrées sur les groupes de Lie à croissance polynomiale du volume, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), 539--542.

Mathematical Reviews (MathSciNet): MR99i:43011

Digital Object Identifier: doi:10.1016/S0764-4442(98)85003-9

--------, Sub-Laplacians with drift on Lie groups of polynomial volume growth, Mem. Amer. Math. Soc. 155 (2002), no. 739.

Mathematical Reviews (MathSciNet): MR2003c:22015

J.-P. Anker, $L_p$ Fourier multipliers on Riemannian symmetric spaces of the noncompact type, Ann. of Math. (2) 132 (1990), 597--628.

Mathematical Reviews (MathSciNet): MR92e:43006

Digital Object Identifier: doi:10.2307/1971430

JSTOR: links.jstor.org

--. --. --. --., The spherical Fourier transform of rapidly decreasing functions. A simple proof of a characterization due to Harish-Chandra, Helgason, Trombi, and Varadarajan, J. Funct. Anal. 96 (1991), 331--349.

Mathematical Reviews (MathSciNet): MR92d:22008

Digital Object Identifier: doi:10.1016/0022-1236(91)90065-D

--. --. --. --., Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Math. J. 65 (1992), 257--297.

Mathematical Reviews (MathSciNet): MR93b:43007

Digital Object Identifier: doi:10.1215/S0012-7094-92-06511-2

Project Euclid: euclid.dmj/1077295136

J.-P. Anker and L. Ji, Heat kernel and Green function estimates on noncompact symmetric spaces, Geom. Funct. Anal. 9 (1999), 1035--1091.

Mathematical Reviews (MathSciNet): MR2001b:58038

Digital Object Identifier: doi:10.1007/s000390050107

A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Stud. Math. Appl. 5, North-Holland, Amsterdam, 1978.

Mathematical Reviews (MathSciNet): MR82h:35001

P. Bougerol, Théorème centrale limite local sur certains groupes de Lie, Ann. Sci. École. Norm. Sup. (4) 14 (1981), 403--432.

Mathematical Reviews (MathSciNet): MR83g:60019

C. Conca and M. Vanninathan, Homogenization of periodic structures via Bloch decomposition, SIAM J. Appl. Math. 57 (1997), 1639--1659.

Mathematical Reviews (MathSciNet): MR98j:35017

Digital Object Identifier: doi:10.1137/S0036139995294743

JSTOR: links.jstor.org

R. Gangolli, Asymptotic behavior of spectra of compact quotients of certain symmetric spaces, Acta Math. 121 (1968), 151--192.

Mathematical Reviews (MathSciNet): MR39:360

Digital Object Identifier: doi:10.1007/BF02391912

S. Helgason, Groups and Geometric Analysis, Pure Appl. Math. 113, Academic Press, Orlando, Fla., 1984.

Mathematical Reviews (MathSciNet): MR86c:22017

V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer, Berlin, 1994.

Mathematical Reviews (MathSciNet): MR96h:35003b

M. Kotani and T. Sunada, Albanese maps and off diagonal long time asymptotics for the heat kernel, Comm. Math. Phys. 209 (2000), 633--670.

Mathematical Reviews (MathSciNet): MR2001h:58036

Digital Object Identifier: doi:10.1007/s002200050033

N. Lohoué, Estimations $L^p$ des coefficients de représentation et opérateurs de convolution, Adv. in Math. 38 (1980), 178--221.

Mathematical Reviews (MathSciNet): MR82m:43004

Digital Object Identifier: doi:10.1016/0001-8708(80)90004-3

--. --. --. --., Inégalités de Sobolev pour les sous-laplaciens de certains groupes unimodulaires, Geom. Funct. Anal. 2 (1992), 394--420.

Mathematical Reviews (MathSciNet): MR94d:58141

Digital Object Identifier: doi:10.1007/BF01896661

N. Lohoué and G. Alexopoulos, On the asymptotic behavior of convolution powers and heat kernels on Lie groups, to appear in Contemp. Math.

Mathematical Reviews (MathSciNet): MR2077028

N. Lohoué and T. Rychener, Some function spaces on symmetric spaces related to convolution operators, J. Funct. Anal. 55 (1984), 200--219.

Mathematical Reviews (MathSciNet): MR85d:22024

Digital Object Identifier: doi:10.1016/0022-1236(84)90010-7

J. H. Ortega and E. Zuazua, Large time behavior in $\mathbbR^N$ for linear parabolic equations with periodic coefficients, Asymptot. Anal. 22 (2000), 51--85.

Mathematical Reviews (MathSciNet): MR2001b:35136

P. Ostellari, Global behavior of the heat kernel associated with certain sub-Laplacians on semisimple Lie groups, J. Funct. Anal. 199 (2003), 521--534. \CMP1 971 905

Mathematical Reviews (MathSciNet): MR1971905

Digital Object Identifier: doi:10.1016/S0022-1236(02)00078-2

E. V. Sevostjanova, An asymptotic expansion of the solution of a second order elliptic equation with periodic coefficients, Math. USSR-Sb. 43, no. 2 (1982), 181--198.

Mathematical Reviews (MathSciNet): MR83d:35038

N. T. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and Geometry on Groups, Cambridge Tracts in Math. 100, Cambridge Univ. Press, 1992.

Mathematical Reviews (MathSciNet): MR95f:43008

G. Warner, Harmonic Analysis on Semi-Simple Lie Groups, I, Grundlehren Math. Wiss. 188, Springer, New York, 1972.

Mathematical Reviews (MathSciNet): MR58:16979

--------, Harmonic Analysis on Semi-Simple Lie Groups, II, Grundlehren Math. Wiss. 189, Springer, New York, 1972.

Mathematical Reviews (MathSciNet): MR58:16980

V. V. Zhikov, Spectral approach to the asymptotic diffusion problems (in Russian), Differentsialnye Uravneniya 25, no. 1 (1989), 44--55., 180; English translation in Differential Equations 25, no. 1 (1989), 33--39.

Mathematical Reviews (MathSciNet): MR90a:35107

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